An entropy function of a quantum channel

Motivated by the isomorphic correspondence between quantum channels and their Choi states, we define an entropy function of a quantum channel by the entropy of its Choi state. We show that it satisfies all axioms of the entropy function. We also define the relative entropy of a quantum channel and give the relation between the entropy and the relative entropy of a quantum channel. Moreover, comparing with the entropy of a quantum channel which is defined by Gour and Wilde, we find that the two definitions of the channel entropy are equal for covariant channels. As examples, we compute the entropies of the erasure channel, the d-dimensional depolarizing channel, and a particular kind of Werner–Holevo channels, respectively.